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The effect of the support on spherical langmuir-probe characteristics
Physicn 72 (1974) 417-430
Q Norfh-Holland Publishing Co.
THE EFFECT
OF THE SUPPORT ON SPHERICAL
LANGMUIR-PROBE S.L.F. RICHARDS*,
CHARACTERISTICS
R.P. JONES and G.J. LLOYD
School of Physics, University of Bath, Bath, Somerset,
England
Received 14 June 1973
Synopsis One of the most common designs of a spherical probe consists of a sphere suspended from a glass support by a short wire stem. The effect of the support on both d.c. and second-derivative spherical-probe characteristics has been investigated. A probe has been constructed in which the length of the stem may be varied. It has been found that at short stem lengths the probe underestimates the plasma electron density. Two possible explanations of this are proposed; that the support may be physically screening the probe, or that the probe is interacting with the ion sheath surrounding the support. Experiments have been conducted in the positive column of a helium discharge (p = 0.2 torr; 0.2 mm < &, < 0.5 mm, ilo being the Debye length) which indicate that the latter effect is predominant. Electron energy distributions, derived from the probe’s second-derivative characteristics, indicate an underestimation of the number of the low-energy electrons in the plasma when the probe is close to its support. This is consistent with an interaction with the ion sheath, and a simple theoretical discussion of the phenomena is presented.
1. Introduction. Many theoretical studies of Langmuir probes have been undertaken using cylindrical and spherical geometries (Swift and Schwarl)). In the cylindrical case, the probe is assumed to be a wire of infinite extent, and end effects of negligible importance. This will only be valid if the probe length (a) greatly exceeds the Debye length (A,,). Experimental observations in a region where this inequality no longer holds have recently been reported by Smith and Plumbz). Considering the spherical case, the probe is assumed to have spherical symmetry and no account is taken of the effects of the probe support and stem (fig. 1). Perturbations introduced by these departures from the ideal situation have led some experimental workers to prefer the use of cylindrical probes, where the perturbation may be estimated from the ratio &/a. Boyd and Twiddy3) have shown that it is preferable to use a spherical probe for measurements in the presence of an anisotropic electron velocity distribution * Present address: Physics Department,
University of British Columbia, Vancouver, Canada, 417
418
S.L. F. RICHARDS,
when using the Druyvesteyn4)
R.P.
analysis.
JONES
AND
In designing
G. J. LLOYD
a probe for use under
these
circumstances two conflicting requirements must be observed: (1) The sphere should be sufficiently removed from its glass support (fig. 1) that it is no longer influenced by the positive ion sheath around the support. This implies a long stem. (2) The stem should be sufficiently pared with that of the sphere.
short that its surface
FpQ support
Fig.
I. A spherical
probe,
showing
area is negligible
com-
Stem
SpherIcal probe
the method
by which
it is supported.
To minimise the perturbation of the plasma by the probe the radius of the sphere (TJ is generally restricted to about 0.25 mm. To retain sufficient mechanical rigidity a minimum stem diameter of 50 pm is used by the authors. Thus, if the area of the stem is to be 10 % of the area of the sphere, the stem length must be restricted to 0.5 mm. Should the Debye length become comparable with the stem length (as may well be the case in, for example a positive column), the perturbation of the probe characteristic by the positive ion sheath surrounding the support will become significant. The only experimental investigation of this possibility known to the authors is that of Sloane and Emeleus5). They employed a variable-length cylindrical probe with a copper support; these were insulated from one another by a coaxial glass shield. The effect on the probe characteristic of varying the potential applied to the copper support was investigated. They found that the application of large negative potentials to the support caused the probe to underestimate the number of low-energy electrons. To extend these observations to the case of a spherical probe. a probe has been constructed in which the stem length may be varied from 7 mm (about 3OA,,) to zero. The characteristic has been investigated in the positive column of a helium discharge where the ratio of the mean electron thermal speed to the electron drift velocity was about 40: 1, thus ensuring a close approach to an isotropic electron velocity distribution. 2. Theory. The theoretical work in this section is concerned with the current to the probe when it is biased to space potential. Space potential is ill-defined at short stem lengths, since the portion of the stem close to the glass support is inside the ion sheath surrounding the support. In this region the local space potential varies rapidly with distance. It will be assumed throughout that the electron mean free path is long enough to enable collisionless probe theories to be employed.
PERTURBATION
OF SPHERICAL
PROBE CHARACTERISTICS
419
The random electron current density (J) in the plasma is given by J = &Fe,
where n is the electron density, e the electronic charge, and E the mean electron speed. The current (lo) to the probe at space potential is the product of the random electron current density and the probe area. The total surface area of the probe may be considered to be the sum of two parts; that due to the sphere (A,) and that due to the stem (A,). Thus we may write: I,, = J(As + A,).
(1)
This ignores the effect of the re-entrant part of the probe surface formed where the stem joins the sphere. Two effects which may cause the probe current to deviate from the value given by (1) are: (a) The screening of the stem and part of the ’ surface of the sphere by the glass support (solid-angle screening). Provided the electron mean free path is sufficiently long, this will only depend upon support geometry. (b) The change in the local electron number density, and space potential, when a significant part of the surface area of the probe enters the ion sheath around the support. Both these effects will be significant only at shorter stem lengths, but the stem length for the onset of (a) is determined by probe geometry, whilst that for (b) is also dependent upon Debye length. These effects will now be considered in greater detail. 2.1. Solid-angle screening. Assuming collisionless, straight-line electron trajectories, eq. (1) may be written to include the effect of solid-angle screening as:
where Fs and I;, are the fractional solid angles of the plasma unscreened by the probe support; in the case of the stem, F, must be some average value. Using the notation of fig. 2, the fractional solid angle at the sphere screened by the support is: f = t
u - p(RZ + /32)-f].
In deriving this expression, the assumption is made that the sphere may be represented by a point at its centre. This is invalid at very short stem lengths. Defining ac, = R//3,Fs may be expressed as: F, = 1 -f
= + [l + (1 + a~:)-+].
(4)
420
S.L. F. RICHARDS,
Fig. 2. The notation
R.P. JONES AND G. J. LLOYD
used in the solid-angle screening theory.
From a point on the surface of the stem, only half the support is visible; the maximum area of plasma accessible is restricted to a hemisphere. element of stem of length dl distant 1 from the support (fig. 2) will have nal solid angle screened given by (3) with 1 replacing p. The average solid angle screened for a stem of length L will be: J; = (l/L) ;,j’d/
= (1/2L) [L + R - (R2 +
however, Thus the a fractiofractional
L2)%].
0
Fig. 3. Variation of the current to the probe at space potential with stem area, as predicted by eq. (I) [curve (a)] and eq. (2) [curve (b)]. The curves do not coalesce at long stem lengths since a portion of the stem is always screened by the support [eq. (5)]. Stem diameter 50 t*m, 1’S = 0.25mm, R = 0.75 mm.
PERTURBATION
OF SPHERICAL PROBE CHARACTERISTICS
421
Defining LY,= R/L, Fc may be expressed as: F, = 1 -fC = -)[l -a,+(1
+&].
(5)
Fig. 3 shows a graph of the current to the probe versus stem area as predicted by these equations, together with (1) for comparison. The ordinate is chosen as Z,/J for convenience and the sphere was taken as 0.5 mm diameter on a 50 pm diameter stem, with a 1.5 mm diameter glass support. Supports of this size have been found necessary experimentally to overcome the effect of capacity coupling first reported by Olson and MedicuP). The curves of fig. 3 do not coalesce at long stem lengths since a portion of the stem remains shielded by the support. In practice, there is a small uncertainty in the numerical value of support radius which should be employed, since the support is surrounded by a positive ion sheath. This sheath will bend the trajectory of any electron entering it; hence an electron which (in the absence of the sheath) would have grazed the support and then been collected by the probe will, on encountering the sheath, be deflected away from the probe. 2.2. Support ion sheath. The glass support (being an insulator) will be at floating potential, negative with respect to the plasma. The support is surrounded by a positive ion sheath; the electron density at a point in the sheath will be less than the density in the plasma. In considering the effect of the sheath on the parts of the probe immersed in it, it will be assumed firstly that the presence of the probe does not significantly disturb the sheath, and secondly that the probe is biased to the space potential in the plasma outside the sheath. As a consequence of the second assumption, the parts of the probe in the sheath will be biased positively with respect to the local “space potential” in the sheath. Two factors must be taken into account when assessing the effect of the sheath on the electron current collected by the parts of the probe immersed in it; the decrease in electron density towards the support, and the change in the local space potential. To estimate the relative importance of these factors, consider a small section of the stem immediately adjacent to the support. Since the probe is biased to the space potential outside the sheath, this section of the stem will be positive with respect to the adjacent sheath region by the numerical value of the floating potential (V,); hence the electron current flowing to it will be increased. In our particular discharge eV,/kT, is typically -2.2. Making the assumption that the characteristic of this section of the stem obeys the classic cylindrical Langmuir orbital-motion limited expression, we may estimate the factor by which the current flowing to this section is increased, as: (1 + leV,/kT,l)i
N 1.79.
The factor by which the current is reduced, due to the decrease in electron number density in this region may be estimated from the work of Andrews and
422
S.L.F. RICHARDS, R.P. JONES AND G. J. LLOYD
Varey7). They predict that the electron number density adjacent to a plane probe at the floating potential stated is about 5 ‘A of its value at infinity. As the decrease in current will be proportional to the decrease in number density, this effect will greatly outweigh that due to changes in space potential. Hence there will be a net decrease in the current per unit length flowing to the section of the stem inside the ion sheath. To obtain more accurate estimates of the variation of the current with stem length, a computer programme was written which calculated the current to each section of the probe in the sheath due to the reduced density of electrons around that section. In view of the large disparity between the two effects, initially the change in space potential was ignored. Two theoretical models were employed for the sheath surrounding the support; that proposed by Andrews and Varey’) and that by Allen et aL8), modified by additional terms to account for the finite collection of electrons by the probe (Swift and Schwar’), ch. 3). Both models predicted currents to the probe of about 40 % of the experimentally observed values. It is estimated that allowance for the change in potential would improve this figure to about 60x, but this has not been incorporated. It is thought that, since the sphere radius is of the same order as the Debye length, the assumption that the sheath is not perturbed by the probe is violated, rendering the calculations invalid. 3. Ekperimental details. 3.1. Discharge tube. The discharge tube was of 8 cm diameter and its length about 70 cm. The tube was constructed of Pyrex glass and employed a 7 cm diameter nickel anode (the reference electrode) and a 50 W tungsten filamentary cathode. The positive column of a helium discharge was used for this investigation and the majority of the measurements were taken at a pressure of 0.1 torr; further observations at a pressure of 0.2 torr showed no significant differences. Discharge currents between 0.02 and 0.3 A were used. Stringent vacuum techniques ensured adequate purity and the contaminant level was monitored by a mass spectrometer. Background pressures of less than 2 x lo-’ torr were recorded prior to the admission of gas to the system. 3.2. Measuring has been described
system. The d.c. and elsewhere’).
second-derivative
measuring
system
3.3. The probe. Fig. 4 shows the design of the variable stem length probe. The glass support is situated in a side arm attached to the discharge tube, and is located by a constriction. Above the constriction is a glass-metal seal and a metal bellows mounted on a flange (for easy access). A mating flange carries another glass-metal seal and a tungsten seal mounted on the axis of the side arm. Thus the whole assembly above the metal bellows can be moved with respect to the discharge tube. The platinum spherical probe and its 50 pm (diameter) stem wire are supported from the tungsten seal by a length of nickel wire. Vertical move-
PERTURBATION
OF SPHERICAL
PROBE CHARACTERISTICS
423
ment of the upper section of the assembly will cause the probe to move within its glass support, thus varying the stem length. The total vertical travel is limited only by the degree of movement of the bellows, in this case 7 mm. f
1b
Tungatcn feed through
u1 q6IM
seal
Clamp -
Clamp ”
‘~G/MSeal Constriction
Tube wall Glass probe support
Fig. 4. The variable stem length probe, showing the method of construction.
A pair of clamps (not shown in fig. 4) are used to grip the assembly, and are located on the lower section of the bellows and the lower part of the flange assembly. The distance between the clamps is controlled by a micrometer screw head, thus permitting the measurement of the change in length of the stem of the probe to an accuracy of +O.Ol mm. Absolute measurement of stem length requires knowledge of the micrometer reading corresponding to zero stem length. This is obtained by raising the probe until the sphere is just touching the glass support, as observed by a microscope. The maximum error in the measurement of stem length is estimated at +O.l mm and corresponds to a 2 % error in the stem area compared with the area of the sphere. The diameter of another sample of the wire used for the stem was measured by both optical wedge and laser interferometric techniques, and was found to be within 0.1 ‘A of its nominal 50 pm. The diameter of the sphere (nominally 0.5 mm) was measured several times prior to insertion into the tube, using a micrometer screw gauge. An average of these values was employed in calculating the area of the sphere, and it is estimated that the maximum error in the area is about 8%. Thus the total area of the probe at any stem length is known to an accuracy of 10% of the area of the sphere. Two glass-support diameters (0.5 mm and 1.5 mm) have been employed in these investigations. General views of the probe with the stem well extended are shown in fig. 5. The hole in the support through which the stem protrudes is sufficiently small to approximate to the seal normally found in this position in fixed stem probes.
424 In operation,
S.L.F. RICHARDS,
the clamps
R.P. JONES AND G.J. LLOYD
and other
sembly are held at the probe floating fields which might perturb the plasma
metalwork potential,
associated thus avoiding
with the probe
as-
any stray electric
-I
Im
Fig. 5. The probes and supports used in the investigation.
(a) 0.25 mm radius support.(b)
0.75mm
radius support.
4. Results. 4.1. Current to probe at space potential. In this investigation space potential was taken to be the point at which the second derivative of the probe characteristic was zero.
PERTURBATION
OF SPHERICAL
PROBE CHARACTERISTICS
425
To compare results for different random electron current densities (J), it is necessary to plot the data as shown in fig. 3 (Z,/J US.A,, A, being the area of stem exposed). This requires a knowledge of J. From eq. (2) it may be seen that a plot of I,, us. A, will be linear when the sphere is remote from its support. Hence .Z may be determined by either: (a) the gradient of the linear portion of I0 tis. A,, or (b) the extrapolated intercept of the line, knowing the area of the sphere (As).
I.6-
. $,=0.23mm
kO.25
0 A,=0.32mm
k0.l
a &=0.45mm
1.6. ’
Stem length-mm 2 3 4
I i
4 St*m
b
A A
I;O.OSA
5 8
6
7 IO
8 I2
area-m*xld
Fig. 6. Results taken with 0.25 mm radius support at a pressure of 0.1 torr and various discharge currents (In). The Debye length (An) is also tabulated. The solid curves above the points are the predictions of the unscreened and solid-angle screened theories. The line through the points is for comparison purposes.
The first method involves smallest error and so a least-squares fit was applied to the data to evaluate the gradient. The values of J obtained were applied to method (b) to give an estimate of A,. In the case of the data shown in fig. 6, this estimate was within 8 % of that obtained by physical measurement. This is within the experimental error of 10% detailed above, and may also be partly explained by the sphere screening the stem. Fig. 6 shows results taken with the 0.25 mm radius glass support shown in fig. 5a. Also shown are the theoretical curves corresponding to eqs. (1) and (2),
426
S.L.F. RICHARDS,
R.P. JONES AND G. J. LLOYD
the unscreened and solid-angle screened cases, respectively. The error in A, accounts for the lack of coincidence between the experimental and theoretical curves. However, if a new value of A, is chosen so that the linear portions align, then the experimental curve deviates from linearity at a greater stem length than that predicted by solid-angle screening. It is the authors’ view that this deviation is caused by the sphere entering the ion sheath surrounding the support. The experimental data discussed cover a range of Debye lengths in the ratio of 2: 1, yet the distance from the support at which the experimental curve deviates from linearity remains constant at about 2 mm. This is evidently contradictory to the concept that the thickness of the positive ion sheath is proportional to the Debye length. To resolve this difficulty numerical computations have been made of the potential profile within the sheath for various R/A,, assuming that the sheath around the end of the support approximates to spherical geometry. These computations were based on the theory of Allen et ~1.~) because experience has shown that, for a probe at floating potential, the change in the potential profiles caused by the finite collection of electrons by the probe is negligible. Fig. 7 shows the results of these computations in the form of a graph of r,/R vs. R/R,, where y0 is the sheath radius. The sheath edge was taken to be the point at which eV/kT, = -0.5. Note that the quantity r,/R is proportional to the physical thickness of the sheath. From fig. 7 it is evident that the thickness of the sheath varies by only 8% over the range of values of R/l,, encountered in this investigation; thus the distance from the support at which the experimental curve deviates from linearity will change by only 8% as compared with the factor of two previously discussed. Another series of experiments was performed with a glass support of radius 0.75 mm. Fig. 8 shows a typical set of results plotted with those for a 0.25 mm radius glass support for comparison. The deviation from linearity still occurs at a stem length of 2 mm, and again this must be ascribed to a sheath effect since the solid-angle screening predicts a shorter stem length. However, in this case, it is felt that the end of the support (fig. 5b) departs so radically from spherical geometry that there would be little value in a computation
of the sheath thickness.
4.2. Electron energy distributions. The theory described in section 2 is based on the assumption that the mean electron energy does not vary in the vicinity of the probe. To ensure that this assumption was valid, the second derivative of the probe characteristic (d21/dVZ) was recorded at various stem lengths. The relationship
first derived by Druyvesteyn4) was employed to derive the electron energy distribution from the measured second derivatives; kTJe was then obtained from the
2.0
1.8 s
A I.6
I.4
‘? 1.2
0.3
0.5
2.0
‘.O Ry
3.0
5.0
I
10.0
D
Fig. 7. Sheath radius (~a) plotted against support radius (R) in Debye lengths. Note that the sheath radius is plotted in units of support radius instead of Debye length. The data were obtained from numerical solutions of the equation due to Allen et ~1.~) for a spherical probe at floating potential.
Fig. 8. Comparison between 0.25 mm (x ) and 0.75 mm (0) radii supports. The measurements were made at a pressure of 0.1 torr, with Debye lengths of 0.32 mm and 0.30 mm, respectively. In both cases the discharge current was 0.1 A.
428
S.L.F. RICHARDS,
R.P. JONES AND G.J. LLOYD
distribution. Fig. 9 shows a typical result; it is evident that kT,/e is constant to ’ within _+5 % of its mean value and hence J was constant to within 2.5%. Fig. 10 shows a complete
set of energy
distributions
plotted
as a function
of
stem length. When the probe is close to the support there is a marked depletion of low-energy electrons, while the effect of the support is less marked with regard to the high-energy electrons. These experimental observations are in agreement with the results reported
by Sloane and Emeleus5).
I
56
5% --b-0 0
5.6.
0
j > 0
lJ
5.45cv > u’54O
0
j.
52
_
avcragc
0
’ ;
i”
0
*
o
o
: 0 0 p, ----_--__-___________ 0
sob
ho: 5.0
51cm 0
I
2
Length
3
4
mm 5
6
7
Fig. 9. Variation of the electron energy (obtained from the energy distribution) with stem length. Data obtained with 0.75 mm radius support at a pressure of 0.2 torr and discharge current 0.2 A.
Electron energy-cV Fig. 10. Variation
of the electron
energy distribution as fig. 9.
with stem length. Discharge
parameters
PERTURBATION
OF SPHERICAL
PROBE CHARACTERISTICS
429
Assuming that the electron velocity distribution remote from the probe has the form .f(cXc,,cZ)dc, dc,, dc, , where the parameters have their usual notation, then electrons moving in the positive x direction will be retarded by the negative potential in the sheath. At some point in the sheath, where the potential is V volts (V being numerically negative), the x component of the electron’s velocity (c,) is given by: c: - c,” = 2eVjm. Hence the velocity distribution function at this point can be expressed as :
cs(2 - 2eV/m)-+f [(c,Z- 2eV/m)‘,
c, , cz]
dc, dc, dc, .
Thus one of the effects of retarding the velocity distribution is that the original function is multiplied by the term c, (ct - 2eV/m)-i. Fig. 11 shows a graph of this function US.c, for various values of V, and it can be seen that for c, small it tends to zero and approaches unity as c, becomes larger. The effect of a retarding potential in an electron velocity distribution is thus to preferentially reduce the number of low-energy electrons in the distribution. This argument is only valid in the absence of collisions in the sheath, which is the case in our investigation. No detailed comparisons with experiment have been made as the distributions measured by the variable stem length probe are not spatially well resolved, being a weighted “average” of the distributions found in the sheath.
Fig. 11. The function C(C2 - &V/m)-* plotted against C. Values of retarding potential of 1.0, 5.0 and 11.0 V have been chosen for graphs (a), (b) and (c), respectively.
(V)
430 From
S. L. F. RICHARDS,
fig. 10, it is apparent
R.P. JONES AND G. J. LLOYD
that the energy
distributions
are substantially
con-
stant for a stem length greater than 2mm. This coincides with the point at which IO/J becomes a linear function of stem length as shown in fig. 8. As the stem length is increased from 2 mm to 7 mm, a small decrease (6% in the total number density) is observed in the number of low-energy electrons in the distribution. Since the tip of the support was located on the tube axis, it is thought that this was due to the radial variation of the electron number density in the column. Kinderdijk and Van Eck’O) have reported decreases in number density of between 5 and 8 y0 over similar fractional discharge-tube radii. 5. Conclusion. The results presented cannot be adequately explained by “physical solid-angle” screening of the probe by the support. We have therefore concluded that the effect is predominantly due to the probe interacting with the ion sheath surrounding the support. Theoretical treatment of this interaction has been discussed, but since the probe significantly distorts the sheath, existing sheath models cannot be utilized. Tt is anticipated that at shorter Debye lengths physical solid-angle screening will become a more significant perturbation. The effect of varying support diameters and stem lengths on the current to the probe at space potential has been experimentally determined. Fig. 8 shows that with a support radius of 0.25 mm, the current to the probe is reduced from its theoretical value by approximately 4 “/, at a stem length of 1.O mm. This is probably the smallest support which may be employed if the capacitive effect discussed by Olson and Medic&) is to be avoided. If a.c. measurements are not used it should be quite feasible to work with a support radius of 0.1 mm. It is estimated that this will make the error negligible down to a stem length of 0.5 mm, enabling the 10 : I sphere/cylinder area ratio to be realized. 6. Acknowledgments. The authors are indebted to the SRC for providing the salary of one of us (SLFR). We would like to express our thanks to Mr. M.D. D. Parsons for his invaluable practical assistance, and to Mr. M.R. Lock for the fabrication of the glassware. REFERENCES 1) Swift, J. D. and Schwar, M. J. R., Electrical Probes for Plasma Diagnostics, Iliffe (London, 1970). 2) Smith, D. and Plumb, I.C., J. Phys. D5 (1972) 1226. 3) Boyd, R.L. F. and Twiddy, N.D., Proc. Roy. Sot. A250 (1959) 53. 4) Druyvesteyn, M. J., Z. Phys. 64 (1930) 781. 5) Sloane, R.H. and Emeleus, K.G., Phys. Rev. 44 (1933) 333. 6) Olson, R.A. and Medicus, G., J. appl. Phys. 38 (1967) 4539. 7) Andrews, J.G. and Varey, R.H., J. Phys. A3 (1970) 413. 8) Allen, J.E., Boyd, R.L.F. and Reynolds, P., Proc. Roy. Sot. 70 (1957) 297. 9) Richards, S. L. F., Lloyd, G. J. and Jones, R.P., J. Phys. E5 (1972) 595. IO) Kinderdijk, H. M. J. and Van Eck, J., Physica 59 (1972) 257.
Q Norfh-Holland Publishing Co.
THE EFFECT
OF THE SUPPORT ON SPHERICAL
LANGMUIR-PROBE S.L.F. RICHARDS*,
CHARACTERISTICS
R.P. JONES and G.J. LLOYD
School of Physics, University of Bath, Bath, Somerset,
England
Received 14 June 1973
Synopsis One of the most common designs of a spherical probe consists of a sphere suspended from a glass support by a short wire stem. The effect of the support on both d.c. and second-derivative spherical-probe characteristics has been investigated. A probe has been constructed in which the length of the stem may be varied. It has been found that at short stem lengths the probe underestimates the plasma electron density. Two possible explanations of this are proposed; that the support may be physically screening the probe, or that the probe is interacting with the ion sheath surrounding the support. Experiments have been conducted in the positive column of a helium discharge (p = 0.2 torr; 0.2 mm < &, < 0.5 mm, ilo being the Debye length) which indicate that the latter effect is predominant. Electron energy distributions, derived from the probe’s second-derivative characteristics, indicate an underestimation of the number of the low-energy electrons in the plasma when the probe is close to its support. This is consistent with an interaction with the ion sheath, and a simple theoretical discussion of the phenomena is presented.
1. Introduction. Many theoretical studies of Langmuir probes have been undertaken using cylindrical and spherical geometries (Swift and Schwarl)). In the cylindrical case, the probe is assumed to be a wire of infinite extent, and end effects of negligible importance. This will only be valid if the probe length (a) greatly exceeds the Debye length (A,,). Experimental observations in a region where this inequality no longer holds have recently been reported by Smith and Plumbz). Considering the spherical case, the probe is assumed to have spherical symmetry and no account is taken of the effects of the probe support and stem (fig. 1). Perturbations introduced by these departures from the ideal situation have led some experimental workers to prefer the use of cylindrical probes, where the perturbation may be estimated from the ratio &/a. Boyd and Twiddy3) have shown that it is preferable to use a spherical probe for measurements in the presence of an anisotropic electron velocity distribution * Present address: Physics Department,
University of British Columbia, Vancouver, Canada, 417
418
S.L. F. RICHARDS,
when using the Druyvesteyn4)
R.P.
analysis.
JONES
AND
In designing
G. J. LLOYD
a probe for use under
these
circumstances two conflicting requirements must be observed: (1) The sphere should be sufficiently removed from its glass support (fig. 1) that it is no longer influenced by the positive ion sheath around the support. This implies a long stem. (2) The stem should be sufficiently pared with that of the sphere.
short that its surface
FpQ support
Fig.
I. A spherical
probe,
showing
area is negligible
com-
Stem
SpherIcal probe
the method
by which
it is supported.
To minimise the perturbation of the plasma by the probe the radius of the sphere (TJ is generally restricted to about 0.25 mm. To retain sufficient mechanical rigidity a minimum stem diameter of 50 pm is used by the authors. Thus, if the area of the stem is to be 10 % of the area of the sphere, the stem length must be restricted to 0.5 mm. Should the Debye length become comparable with the stem length (as may well be the case in, for example a positive column), the perturbation of the probe characteristic by the positive ion sheath surrounding the support will become significant. The only experimental investigation of this possibility known to the authors is that of Sloane and Emeleus5). They employed a variable-length cylindrical probe with a copper support; these were insulated from one another by a coaxial glass shield. The effect on the probe characteristic of varying the potential applied to the copper support was investigated. They found that the application of large negative potentials to the support caused the probe to underestimate the number of low-energy electrons. To extend these observations to the case of a spherical probe. a probe has been constructed in which the stem length may be varied from 7 mm (about 3OA,,) to zero. The characteristic has been investigated in the positive column of a helium discharge where the ratio of the mean electron thermal speed to the electron drift velocity was about 40: 1, thus ensuring a close approach to an isotropic electron velocity distribution. 2. Theory. The theoretical work in this section is concerned with the current to the probe when it is biased to space potential. Space potential is ill-defined at short stem lengths, since the portion of the stem close to the glass support is inside the ion sheath surrounding the support. In this region the local space potential varies rapidly with distance. It will be assumed throughout that the electron mean free path is long enough to enable collisionless probe theories to be employed.
PERTURBATION
OF SPHERICAL
PROBE CHARACTERISTICS
419
The random electron current density (J) in the plasma is given by J = &Fe,
where n is the electron density, e the electronic charge, and E the mean electron speed. The current (lo) to the probe at space potential is the product of the random electron current density and the probe area. The total surface area of the probe may be considered to be the sum of two parts; that due to the sphere (A,) and that due to the stem (A,). Thus we may write: I,, = J(As + A,).
(1)
This ignores the effect of the re-entrant part of the probe surface formed where the stem joins the sphere. Two effects which may cause the probe current to deviate from the value given by (1) are: (a) The screening of the stem and part of the ’ surface of the sphere by the glass support (solid-angle screening). Provided the electron mean free path is sufficiently long, this will only depend upon support geometry. (b) The change in the local electron number density, and space potential, when a significant part of the surface area of the probe enters the ion sheath around the support. Both these effects will be significant only at shorter stem lengths, but the stem length for the onset of (a) is determined by probe geometry, whilst that for (b) is also dependent upon Debye length. These effects will now be considered in greater detail. 2.1. Solid-angle screening. Assuming collisionless, straight-line electron trajectories, eq. (1) may be written to include the effect of solid-angle screening as:
where Fs and I;, are the fractional solid angles of the plasma unscreened by the probe support; in the case of the stem, F, must be some average value. Using the notation of fig. 2, the fractional solid angle at the sphere screened by the support is: f = t
u - p(RZ + /32)-f].
In deriving this expression, the assumption is made that the sphere may be represented by a point at its centre. This is invalid at very short stem lengths. Defining ac, = R//3,Fs may be expressed as: F, = 1 -f
= + [l + (1 + a~:)-+].
(4)
420
S.L. F. RICHARDS,
Fig. 2. The notation
R.P. JONES AND G. J. LLOYD
used in the solid-angle screening theory.
From a point on the surface of the stem, only half the support is visible; the maximum area of plasma accessible is restricted to a hemisphere. element of stem of length dl distant 1 from the support (fig. 2) will have nal solid angle screened given by (3) with 1 replacing p. The average solid angle screened for a stem of length L will be: J; = (l/L) ;,j’d/
= (1/2L) [L + R - (R2 +
however, Thus the a fractiofractional
L2)%].
0
Fig. 3. Variation of the current to the probe at space potential with stem area, as predicted by eq. (I) [curve (a)] and eq. (2) [curve (b)]. The curves do not coalesce at long stem lengths since a portion of the stem is always screened by the support [eq. (5)]. Stem diameter 50 t*m, 1’S = 0.25mm, R = 0.75 mm.
PERTURBATION
OF SPHERICAL PROBE CHARACTERISTICS
421
Defining LY,= R/L, Fc may be expressed as: F, = 1 -fC = -)[l -a,+(1
+&].
(5)
Fig. 3 shows a graph of the current to the probe versus stem area as predicted by these equations, together with (1) for comparison. The ordinate is chosen as Z,/J for convenience and the sphere was taken as 0.5 mm diameter on a 50 pm diameter stem, with a 1.5 mm diameter glass support. Supports of this size have been found necessary experimentally to overcome the effect of capacity coupling first reported by Olson and MedicuP). The curves of fig. 3 do not coalesce at long stem lengths since a portion of the stem remains shielded by the support. In practice, there is a small uncertainty in the numerical value of support radius which should be employed, since the support is surrounded by a positive ion sheath. This sheath will bend the trajectory of any electron entering it; hence an electron which (in the absence of the sheath) would have grazed the support and then been collected by the probe will, on encountering the sheath, be deflected away from the probe. 2.2. Support ion sheath. The glass support (being an insulator) will be at floating potential, negative with respect to the plasma. The support is surrounded by a positive ion sheath; the electron density at a point in the sheath will be less than the density in the plasma. In considering the effect of the sheath on the parts of the probe immersed in it, it will be assumed firstly that the presence of the probe does not significantly disturb the sheath, and secondly that the probe is biased to the space potential in the plasma outside the sheath. As a consequence of the second assumption, the parts of the probe in the sheath will be biased positively with respect to the local “space potential” in the sheath. Two factors must be taken into account when assessing the effect of the sheath on the electron current collected by the parts of the probe immersed in it; the decrease in electron density towards the support, and the change in the local space potential. To estimate the relative importance of these factors, consider a small section of the stem immediately adjacent to the support. Since the probe is biased to the space potential outside the sheath, this section of the stem will be positive with respect to the adjacent sheath region by the numerical value of the floating potential (V,); hence the electron current flowing to it will be increased. In our particular discharge eV,/kT, is typically -2.2. Making the assumption that the characteristic of this section of the stem obeys the classic cylindrical Langmuir orbital-motion limited expression, we may estimate the factor by which the current flowing to this section is increased, as: (1 + leV,/kT,l)i
N 1.79.
The factor by which the current is reduced, due to the decrease in electron number density in this region may be estimated from the work of Andrews and
422
S.L.F. RICHARDS, R.P. JONES AND G. J. LLOYD
Varey7). They predict that the electron number density adjacent to a plane probe at the floating potential stated is about 5 ‘A of its value at infinity. As the decrease in current will be proportional to the decrease in number density, this effect will greatly outweigh that due to changes in space potential. Hence there will be a net decrease in the current per unit length flowing to the section of the stem inside the ion sheath. To obtain more accurate estimates of the variation of the current with stem length, a computer programme was written which calculated the current to each section of the probe in the sheath due to the reduced density of electrons around that section. In view of the large disparity between the two effects, initially the change in space potential was ignored. Two theoretical models were employed for the sheath surrounding the support; that proposed by Andrews and Varey’) and that by Allen et aL8), modified by additional terms to account for the finite collection of electrons by the probe (Swift and Schwar’), ch. 3). Both models predicted currents to the probe of about 40 % of the experimentally observed values. It is estimated that allowance for the change in potential would improve this figure to about 60x, but this has not been incorporated. It is thought that, since the sphere radius is of the same order as the Debye length, the assumption that the sheath is not perturbed by the probe is violated, rendering the calculations invalid. 3. Ekperimental details. 3.1. Discharge tube. The discharge tube was of 8 cm diameter and its length about 70 cm. The tube was constructed of Pyrex glass and employed a 7 cm diameter nickel anode (the reference electrode) and a 50 W tungsten filamentary cathode. The positive column of a helium discharge was used for this investigation and the majority of the measurements were taken at a pressure of 0.1 torr; further observations at a pressure of 0.2 torr showed no significant differences. Discharge currents between 0.02 and 0.3 A were used. Stringent vacuum techniques ensured adequate purity and the contaminant level was monitored by a mass spectrometer. Background pressures of less than 2 x lo-’ torr were recorded prior to the admission of gas to the system. 3.2. Measuring has been described
system. The d.c. and elsewhere’).
second-derivative
measuring
system
3.3. The probe. Fig. 4 shows the design of the variable stem length probe. The glass support is situated in a side arm attached to the discharge tube, and is located by a constriction. Above the constriction is a glass-metal seal and a metal bellows mounted on a flange (for easy access). A mating flange carries another glass-metal seal and a tungsten seal mounted on the axis of the side arm. Thus the whole assembly above the metal bellows can be moved with respect to the discharge tube. The platinum spherical probe and its 50 pm (diameter) stem wire are supported from the tungsten seal by a length of nickel wire. Vertical move-
PERTURBATION
OF SPHERICAL
PROBE CHARACTERISTICS
423
ment of the upper section of the assembly will cause the probe to move within its glass support, thus varying the stem length. The total vertical travel is limited only by the degree of movement of the bellows, in this case 7 mm. f
1b
Tungatcn feed through
u1 q6IM
seal
Clamp -
Clamp ”
‘~G/MSeal Constriction
Tube wall Glass probe support
Fig. 4. The variable stem length probe, showing the method of construction.
A pair of clamps (not shown in fig. 4) are used to grip the assembly, and are located on the lower section of the bellows and the lower part of the flange assembly. The distance between the clamps is controlled by a micrometer screw head, thus permitting the measurement of the change in length of the stem of the probe to an accuracy of +O.Ol mm. Absolute measurement of stem length requires knowledge of the micrometer reading corresponding to zero stem length. This is obtained by raising the probe until the sphere is just touching the glass support, as observed by a microscope. The maximum error in the measurement of stem length is estimated at +O.l mm and corresponds to a 2 % error in the stem area compared with the area of the sphere. The diameter of another sample of the wire used for the stem was measured by both optical wedge and laser interferometric techniques, and was found to be within 0.1 ‘A of its nominal 50 pm. The diameter of the sphere (nominally 0.5 mm) was measured several times prior to insertion into the tube, using a micrometer screw gauge. An average of these values was employed in calculating the area of the sphere, and it is estimated that the maximum error in the area is about 8%. Thus the total area of the probe at any stem length is known to an accuracy of 10% of the area of the sphere. Two glass-support diameters (0.5 mm and 1.5 mm) have been employed in these investigations. General views of the probe with the stem well extended are shown in fig. 5. The hole in the support through which the stem protrudes is sufficiently small to approximate to the seal normally found in this position in fixed stem probes.
424 In operation,
S.L.F. RICHARDS,
the clamps
R.P. JONES AND G.J. LLOYD
and other
sembly are held at the probe floating fields which might perturb the plasma
metalwork potential,
associated thus avoiding
with the probe
as-
any stray electric
-I
Im
Fig. 5. The probes and supports used in the investigation.
(a) 0.25 mm radius support.(b)
0.75mm
radius support.
4. Results. 4.1. Current to probe at space potential. In this investigation space potential was taken to be the point at which the second derivative of the probe characteristic was zero.
PERTURBATION
OF SPHERICAL
PROBE CHARACTERISTICS
425
To compare results for different random electron current densities (J), it is necessary to plot the data as shown in fig. 3 (Z,/J US.A,, A, being the area of stem exposed). This requires a knowledge of J. From eq. (2) it may be seen that a plot of I,, us. A, will be linear when the sphere is remote from its support. Hence .Z may be determined by either: (a) the gradient of the linear portion of I0 tis. A,, or (b) the extrapolated intercept of the line, knowing the area of the sphere (As).
I.6-
. $,=0.23mm
kO.25
0 A,=0.32mm
k0.l
a &=0.45mm
1.6. ’
Stem length-mm 2 3 4
I i
4 St*m
b
A A
I;O.OSA
5 8
6
7 IO
8 I2
area-m*xld
Fig. 6. Results taken with 0.25 mm radius support at a pressure of 0.1 torr and various discharge currents (In). The Debye length (An) is also tabulated. The solid curves above the points are the predictions of the unscreened and solid-angle screened theories. The line through the points is for comparison purposes.
The first method involves smallest error and so a least-squares fit was applied to the data to evaluate the gradient. The values of J obtained were applied to method (b) to give an estimate of A,. In the case of the data shown in fig. 6, this estimate was within 8 % of that obtained by physical measurement. This is within the experimental error of 10% detailed above, and may also be partly explained by the sphere screening the stem. Fig. 6 shows results taken with the 0.25 mm radius glass support shown in fig. 5a. Also shown are the theoretical curves corresponding to eqs. (1) and (2),
426
S.L.F. RICHARDS,
R.P. JONES AND G. J. LLOYD
the unscreened and solid-angle screened cases, respectively. The error in A, accounts for the lack of coincidence between the experimental and theoretical curves. However, if a new value of A, is chosen so that the linear portions align, then the experimental curve deviates from linearity at a greater stem length than that predicted by solid-angle screening. It is the authors’ view that this deviation is caused by the sphere entering the ion sheath surrounding the support. The experimental data discussed cover a range of Debye lengths in the ratio of 2: 1, yet the distance from the support at which the experimental curve deviates from linearity remains constant at about 2 mm. This is evidently contradictory to the concept that the thickness of the positive ion sheath is proportional to the Debye length. To resolve this difficulty numerical computations have been made of the potential profile within the sheath for various R/A,, assuming that the sheath around the end of the support approximates to spherical geometry. These computations were based on the theory of Allen et ~1.~) because experience has shown that, for a probe at floating potential, the change in the potential profiles caused by the finite collection of electrons by the probe is negligible. Fig. 7 shows the results of these computations in the form of a graph of r,/R vs. R/R,, where y0 is the sheath radius. The sheath edge was taken to be the point at which eV/kT, = -0.5. Note that the quantity r,/R is proportional to the physical thickness of the sheath. From fig. 7 it is evident that the thickness of the sheath varies by only 8% over the range of values of R/l,, encountered in this investigation; thus the distance from the support at which the experimental curve deviates from linearity will change by only 8% as compared with the factor of two previously discussed. Another series of experiments was performed with a glass support of radius 0.75 mm. Fig. 8 shows a typical set of results plotted with those for a 0.25 mm radius glass support for comparison. The deviation from linearity still occurs at a stem length of 2 mm, and again this must be ascribed to a sheath effect since the solid-angle screening predicts a shorter stem length. However, in this case, it is felt that the end of the support (fig. 5b) departs so radically from spherical geometry that there would be little value in a computation
of the sheath thickness.
4.2. Electron energy distributions. The theory described in section 2 is based on the assumption that the mean electron energy does not vary in the vicinity of the probe. To ensure that this assumption was valid, the second derivative of the probe characteristic (d21/dVZ) was recorded at various stem lengths. The relationship
first derived by Druyvesteyn4) was employed to derive the electron energy distribution from the measured second derivatives; kTJe was then obtained from the
2.0
1.8 s
A I.6
I.4
‘? 1.2
0.3
0.5
2.0
‘.O Ry
3.0
5.0
I
10.0
D
Fig. 7. Sheath radius (~a) plotted against support radius (R) in Debye lengths. Note that the sheath radius is plotted in units of support radius instead of Debye length. The data were obtained from numerical solutions of the equation due to Allen et ~1.~) for a spherical probe at floating potential.
Fig. 8. Comparison between 0.25 mm (x ) and 0.75 mm (0) radii supports. The measurements were made at a pressure of 0.1 torr, with Debye lengths of 0.32 mm and 0.30 mm, respectively. In both cases the discharge current was 0.1 A.
428
S.L.F. RICHARDS,
R.P. JONES AND G.J. LLOYD
distribution. Fig. 9 shows a typical result; it is evident that kT,/e is constant to ’ within _+5 % of its mean value and hence J was constant to within 2.5%. Fig. 10 shows a complete
set of energy
distributions
plotted
as a function
of
stem length. When the probe is close to the support there is a marked depletion of low-energy electrons, while the effect of the support is less marked with regard to the high-energy electrons. These experimental observations are in agreement with the results reported
by Sloane and Emeleus5).
I
56
5% --b-0 0
5.6.
0
j > 0
lJ
5.45cv > u’54O
0
j.
52
_
avcragc
0
’ ;
i”
0
*
o
o
: 0 0 p, ----_--__-___________ 0
sob
ho: 5.0
51cm 0
I
2
Length
3
4
mm 5
6
7
Fig. 9. Variation of the electron energy (obtained from the energy distribution) with stem length. Data obtained with 0.75 mm radius support at a pressure of 0.2 torr and discharge current 0.2 A.
Electron energy-cV Fig. 10. Variation
of the electron
energy distribution as fig. 9.
with stem length. Discharge
parameters
PERTURBATION
OF SPHERICAL
PROBE CHARACTERISTICS
429
Assuming that the electron velocity distribution remote from the probe has the form .f(cXc,,cZ)dc, dc,, dc, , where the parameters have their usual notation, then electrons moving in the positive x direction will be retarded by the negative potential in the sheath. At some point in the sheath, where the potential is V volts (V being numerically negative), the x component of the electron’s velocity (c,) is given by: c: - c,” = 2eVjm. Hence the velocity distribution function at this point can be expressed as :
cs(2 - 2eV/m)-+f [(c,Z- 2eV/m)‘,
c, , cz]
dc, dc, dc, .
Thus one of the effects of retarding the velocity distribution is that the original function is multiplied by the term c, (ct - 2eV/m)-i. Fig. 11 shows a graph of this function US.c, for various values of V, and it can be seen that for c, small it tends to zero and approaches unity as c, becomes larger. The effect of a retarding potential in an electron velocity distribution is thus to preferentially reduce the number of low-energy electrons in the distribution. This argument is only valid in the absence of collisions in the sheath, which is the case in our investigation. No detailed comparisons with experiment have been made as the distributions measured by the variable stem length probe are not spatially well resolved, being a weighted “average” of the distributions found in the sheath.
Fig. 11. The function C(C2 - &V/m)-* plotted against C. Values of retarding potential of 1.0, 5.0 and 11.0 V have been chosen for graphs (a), (b) and (c), respectively.
(V)
430 From
S. L. F. RICHARDS,
fig. 10, it is apparent
R.P. JONES AND G. J. LLOYD
that the energy
distributions
are substantially
con-
stant for a stem length greater than 2mm. This coincides with the point at which IO/J becomes a linear function of stem length as shown in fig. 8. As the stem length is increased from 2 mm to 7 mm, a small decrease (6% in the total number density) is observed in the number of low-energy electrons in the distribution. Since the tip of the support was located on the tube axis, it is thought that this was due to the radial variation of the electron number density in the column. Kinderdijk and Van Eck’O) have reported decreases in number density of between 5 and 8 y0 over similar fractional discharge-tube radii. 5. Conclusion. The results presented cannot be adequately explained by “physical solid-angle” screening of the probe by the support. We have therefore concluded that the effect is predominantly due to the probe interacting with the ion sheath surrounding the support. Theoretical treatment of this interaction has been discussed, but since the probe significantly distorts the sheath, existing sheath models cannot be utilized. Tt is anticipated that at shorter Debye lengths physical solid-angle screening will become a more significant perturbation. The effect of varying support diameters and stem lengths on the current to the probe at space potential has been experimentally determined. Fig. 8 shows that with a support radius of 0.25 mm, the current to the probe is reduced from its theoretical value by approximately 4 “/, at a stem length of 1.O mm. This is probably the smallest support which may be employed if the capacitive effect discussed by Olson and Medic&) is to be avoided. If a.c. measurements are not used it should be quite feasible to work with a support radius of 0.1 mm. It is estimated that this will make the error negligible down to a stem length of 0.5 mm, enabling the 10 : I sphere/cylinder area ratio to be realized. 6. Acknowledgments. The authors are indebted to the SRC for providing the salary of one of us (SLFR). We would like to express our thanks to Mr. M.D. D. Parsons for his invaluable practical assistance, and to Mr. M.R. Lock for the fabrication of the glassware. REFERENCES 1) Swift, J. D. and Schwar, M. J. R., Electrical Probes for Plasma Diagnostics, Iliffe (London, 1970). 2) Smith, D. and Plumb, I.C., J. Phys. D5 (1972) 1226. 3) Boyd, R.L. F. and Twiddy, N.D., Proc. Roy. Sot. A250 (1959) 53. 4) Druyvesteyn, M. J., Z. Phys. 64 (1930) 781. 5) Sloane, R.H. and Emeleus, K.G., Phys. Rev. 44 (1933) 333. 6) Olson, R.A. and Medicus, G., J. appl. Phys. 38 (1967) 4539. 7) Andrews, J.G. and Varey, R.H., J. Phys. A3 (1970) 413. 8) Allen, J.E., Boyd, R.L.F. and Reynolds, P., Proc. Roy. Sot. 70 (1957) 297. 9) Richards, S. L. F., Lloyd, G. J. and Jones, R.P., J. Phys. E5 (1972) 595. IO) Kinderdijk, H. M. J. and Van Eck, J., Physica 59 (1972) 257.